Difference between revisions of "Unramified ideal"
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− | and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u0957908.png" />. More accurately, such an ideal is called absolutely unramified. In general, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u0957909.png" /> be a [[Dedekind ring|Dedekind ring]] with field of fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579010.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579011.png" /> be a finite extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579012.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579013.png" /> be the integral closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579014.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579015.png" /> (cf. [[Integral extension of a ring|Integral extension of a ring]]). A prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579016.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579017.png" /> lying over an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579019.png" /> is unramified in the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579020.png" /> if | + | and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u0957908.png" />. More accurately, such an ideal is called absolutely unramified. In general, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u0957909.png" /> be a [[Dedekind ring|Dedekind ring]] with [[field of fractions]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579010.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579011.png" /> be a finite extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579012.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579013.png" /> be the integral closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579014.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579015.png" /> (cf. [[Integral extension of a ring|Integral extension of a ring]]). A prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579016.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579017.png" /> lying over an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579019.png" /> is unramified in the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579020.png" /> if |
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579021.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095790/u09579021.png" /></td> </tr></table> |
Revision as of 21:20, 28 November 2014
A prime ideal of an algebraic number field
(cf. also Algebraic number; Number field) lying over a prime number
such that the principal ideal
has in
a product decomposition into prime ideals of the form
![]() |
where
![]() |
and . More accurately, such an ideal is called absolutely unramified. In general, let
be a Dedekind ring with field of fractions
, let
be a finite extension of
and let
be the integral closure of
in
(cf. Integral extension of a ring). A prime ideal
of
lying over an ideal
of
is unramified in the extension
if
![]() |
where are pairwise distinct prime ideals of
,
and
. If all ideals
are unramified, then one occasionally says that
remains unramified in
. For a Galois extension
, an ideal
of
is unramified if and only if the decomposition group of
in the Galois group
is the same as the Galois group of the extension of the residue class field
. In any finite extension of algebraic number fields all ideals except finitely many are unramified.
References
[1] | Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966) |
[2] | S. Lang, "Algebraic number theory" , Addison-Wesley (1970) |
[3] | J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1986) |
Unramified ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unramified_ideal&oldid=14372